A295224 -id:A295224 - cp's OEIS frontend

A303864 Array read by antidiagonals: T(n,k) = number of noncrossing path sets on k*n nodes up to rotation with each path having exactly k nodes.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 6, 2, 1, 1, 4, 36, 38, 3, 1, 1, 10, 210, 960, 384, 6, 1, 1, 16, 1176, 18680, 35956, 4425, 14, 1, 1, 36, 6328, 313664, 2280910, 1588192, 57976, 34, 1, 1, 64, 32896, 4683168, 111925464, 323840016, 77381016, 807318, 95, 1

Offset: 0

Andrew Howroyd, May 01 2018

			Array begins:
=======================================================
n\k| 1  2     3        4           5              6
---+---------------------------------------------------
0  | 1  1     1        1           1              1 ...
1  | 1  1     1        3           4             10 ...
2  | 1  1     6       36         210           1176 ...
3  | 1  2    38      960       18680         313664 ...
4  | 1  3   384    35956     2280910      111925464 ...
5  | 1  6  4425  1588192   323840016    46552781760 ...
6  | 1 14 57976 77381016 50668922540 21346459738384 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Columns 2..4 are A002995(n+1), A303865, A303866.
Row n=1 is A051437(k-3).
Cf. A302828, A303844, A303869.
Cf. A295224 (polygon dissections), A303694 (sets of cycles instead of paths).

nmax = 10; seq[n_, k_] := Module[{p, q, h}, p = 1 + InverseSeries[ x/(k*2^If[k == 1, 0, k - 3]*(1 + x)^k) + O[x]^n, x ]; h = p /. x -> x^2 + O[x]^n; q = x*D[p, x]/p; Integrate[((p - 1)/k + Sum[EulerPhi[d]*(q /. x -> x^d + O[x]^n), {d, 2, n}])/x, x] + If[OddQ[k], 0, 2^(k/2 - 2)*x*h^(k/2)] + 1];
Clear[col]; col[k_] := col[k] = CoefficientList[seq[nmax, k], x];
T[n_, k_] := col[k][[n + 1]];
Table[T[n - k, k], {n, 0, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jul 04 2018, after Andrew Howroyd *)

seq(n,k)={ \\ gives gf of k'th column
my(p=1 + serreverse( x/(k*2^if(k==1, 0, k-3)*(1 + x)^k) + O(x*x^n) ));
my(h=subst(p,x,x^2+O(x*x^n)), q=x*deriv(p)/p);
intformal( ((p-1)/k + sum(d=2,n,eulerphi(d)*subst(q,x,x^d+O(x*x^n))))/x) + if(k%2, 0, 2^(k/2-2)*x*h^(k/2)) + 1;
}
Mat(vector(6, k, Col(seq(7, k))))

A221184 Number of colored quivers in the 4-mutation class of a quiver of Dynkin type A_n.

Original entry on oeis.org

1, 1, 3, 19, 118, 931, 7756, 68685, 630465, 5966610, 57805410, 571178751, 5737638778, 58455577800, 602859152496, 6283968796705, 66119469155523, 701526880303315, 7498841128986109, 80696081185766970, 873654669882574580

Offset: 0

N. J. A. Sloane, Jan 22 2013

Also, number of nonequivalent dissections of a polygon into n+1 hexagons by nonintersecting diagonals up to rotation. - Andrew Howroyd, Nov 20 2017

Number of oriented polyominoes composed of n+1 hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. For oriented polyominoes, chiral pairs are counted as two. - Robert A. Russell, Jan 23 2024

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
Hermund A. Torkildsen, Colored quivers of type A and the cell-growth problem, arXiv:1004.4512 [math.RT], 2010.
Hermund A. Torkildsen, Colored quivers of type A and the cell-growth problem, J. Algebra and Applications, 12 (2013), #1250133.

Column k=6 of A295224.

Polyominoes: A004127 (unoriented), A369473 (chiral), A143546 (achiral), A001683(n+2) {3,oo}, A005034 {4,oo}, A005038 {5,oo}.

u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
T[n_, k_] := u[n, k, 1] + (If[EvenQ[n], u[n/2, k, 1], 0] - u[n, k, 2])/2 + DivisorSum[GCD[n - 1, k], EulerPhi[#]*u[(n - 1)/#, k, k/#] &]/k;
a[n_] := T[n + 1, 6];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)
p=6; Table[Binomial[(p-1)n,n]/(((p-2)n+1)((p-2)n+2))+If[OddQ[n],0,Binomial[(p-1)n/2,n/2]/((p-2)n+2)]+DivisorSum[GCD[p,n-1],EulerPhi[#]Binomial[((p-1)n+1)/#,(n-1)/#]/((p-1)n+1)&,#>1&],{n,30}] (* Robert A. Russell, Jan 23 2024 *)

a(n) ~ 5^(5*n + 11/2) / (sqrt(Pi) * n^(5/2) * 2^(8*n + 27/2)). - Vaclav Kotesovec, Jun 15 2018

a(n-1) = A004127(n) + A369473(n) = 2*A004127(n) - A143546(n) = 2*A369473(n) + A143546(n). - Robert A. Russell, Jan 23 2024

a(0)=1 and a(18)-a(20) corrected by Andrew Howroyd, Nov 20 2017

A295495 Number of dissections of an n-gon by nonintersecting diagonals into polygons with a prime number of sides counted up to rotations.

Original entry on oeis.org

1, 1, 2, 5, 11, 36, 114, 410, 1458, 5488, 20786, 80770, 317378, 1265139, 5094139, 20718347, 84961256, 351086326, 1460591637, 6113826319, 25733864299, 108867782794, 462707558813, 1974991841442, 8463121111860, 36397780088126, 157066702354947, 679917566925030

Offset: 3

Andrew Howroyd, Nov 22 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..200
E. Krasko, A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.

Cf. A003455, A295224, A295419.

DissectionsModCyclic[v_] :=
Module[{n = Length[v], q, vars, u, p}, q = Table[0, {n}]; q[[1]] = InverseSeries[x - Sum[x^i v[[i]], {i, 3, Length[v]}]/x + O[x]^(n+1)]; For[i = 2, i <= n, i++, q[[i]] = q[[i-1]] q[[1]]]; vars = Variables[q[[1]]]; u[m_, r_] := Normal[(q[[r]] + O[x]^(Quotient[n, m] + 1))] /. Thread[vars -> vars^m]; p = O[x]^n + x u[1, 1] - x^2 + (u[2, 1] - u[1, 2])/2 + Sum[v[[i]] Sum[EulerPhi[d] u[d, i/d]/i, {d, Divisors[i]}], {i, 3, Length[v]}]; Drop[CoefficientList[p, x], 3]];
DissectionsModCyclic[Boole[PrimeQ[#]]& /@ Range[1, 31]] (* Jean-François Alcover, Sep 26 2019, after Andrew Howroyd *)

\\ number of dissections into parts defined by set.
DissectionsModCyclic(v)={my(n=#v);
my(q=vector(n)); q[1]=serreverse(x-sum(i=3, #v, x^i*v[i])/x + O(x*x^n));
for(i=2, n, q[i]=q[i-1]*q[1]);
my(vars=variables(q[1]));
my(u(m, r)=substvec(q[r]+O(x^(n\m+1)), vars, apply(t->t^m, vars)));
my(p=O(x*x^n) + x*u(1,1) - x^2 + (u(2,1)-u(1,2))/2 + sum(i=3, #v, my(c=v[i]); if(c,c*sumdiv(i, d, eulerphi(d)*u(d,i/d))/i)));
vector(n, i, polcoeff(p, i))}
DissectionsModCyclic(apply(i->isprime(i), [1..30]))

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A303864 Array read by antidiagonals: T(n,k) = number of noncrossing path sets on k*n nodes up to rotation with each path having exactly k nodes.

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A221184 Number of colored quivers in the 4-mutation class of a quiver of Dynkin type A_n.

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A295495 Number of dissections of an n-gon by nonintersecting diagonals into polygons with a prime number of sides counted up to rotations.

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